Non-uniqueness conjecture for s-deep Zero Linear Recurrence Sequences

Let $\{G_n\}_{n=1}^{\infty}$ be an $s$-deep Zero Linear Recurrence Sequence such that $c_{s+1}>s$.

Non-uniqueness conjecture. Uniqueness of decomposition is lost for at least one positive integer $N$.

This conjecture concerns when legal decompositions associated with homogeneous linear recurrences fail to be unique. The paper presents it as an empirical conjecture alongside a theorem proving the same conclusion under the additional hypotheses $c_{s+2}>0$ and $c_L>1$; the stated conjecture remains open in the supplied text.